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' Figure 3.
OPTIMIZING POINT NETWORKS FOR CLOSE-RANGE PHOTOGRAMMETRY:
FIRST TEST RESULTS
E.Tung **, J. Friedrich", F. Karsli*, E. Ayhan"
* KTU, Department of Geodesy and Photogrammetry, 61080 Trabzon, Turkey — (etunc, jurgen, fkarsli, eayhan)@ktu.edu.tr
Commission V, WG V/ 5
KEYWORDS: Optimization, point networks, close-range photogrammetry, law of error propagation
ABSTRACT:
The goal of this paper is to introduce a simple method for optimizing point networks for close-range photogrammetry. The classical
3D formulas are used to derive least square error estimates for a priori unknown field point coordinates X, Y, Z according to the law
of error propagation. These estimates are dependent on assumed errors for all input parameters, such as 3D camera position and
orientation errors as well as uncertainties of a camera's inner orientation. So far, the derived 3D formulas have been tested in the
following ways. Firstly, the analytically gained derivatives were compared with numerically computed ones to ensure their
correctness and secondly, the 3D formulas were checked with the corresponding 2D formulas. Results are positive and it is planned
to implement the derived formulas in an interactive graphical user interface.
1. Introduction
Close-range photogrammetry includes many different
applications like those for the documentation and restoration of
historical buildings. For such a project, a simple method has
been sought to optimize point networks under unchangeable
restrictions like the minimum and maximum camera distances
to objects. Especially the required number of pictures and stereo
models should be minimized as long as 3D coordinate errors
still remain above a given limit.
Among the many different approaches to this problem, it was
assumed that the classical 3D photogrammetric formulas in
combination with the law of error propagation (Koch 1988) lead
to a simple and effective method to fulfill the given
requirements described before.
2. Mathematical Model
The starting point for the mathematical model considered here
are the classical 3D photogrammetric formulas, which can be
written as follows when neglecting the rotation matrix (Kraus
1993):
x-X» Y-X.7-Z
bx. : b'y mi b'z
X-X, Y-Y, Z-Z
bx b"y - bz
where
L-for Ist camera
(I
2 - for 2nd camera
b'(x. y, Z) = R' *(x', y',—c") ;
b'(x,y.2) = R" *(x'', y'',—c'') ;
with
X, Y, Z = 3D coordinates of object points,
Xo.Yo, Zo = 3D coordinates of a camera’s focal point,
R', R"' = Rotation matrices from a camera system to a
local coordinate system,
c — Focal length of a camera, and
x', y' - 2D image coordinates of object points.
The solution of equations (1) for the object coordinates X, Y
and Z gives:
b'y
b'x
*u Yum
b'y by
bx b'x
ny
Ao Y,
b'"x = <
x eX A *X,
b'y b''y
Y= (3)
b'x hx
b'y b"y
Pre M eX,
Z- b'z hz :
b'x GE
bz b'z
The partial derivatives of equations (3) for the given parameters
is given by:
OX, Ax TB; 0X; A,-B,’
X. c] ex 10 3)
OY, Ax Bg 0Y, Ay,-By
ey 1 ay 1
0%, A -B "0X, A -B
gx A, XxX -B, (4)
0Y, A,-B,'0Y, A,-B,
—519—
